[Antreas P. Hatzipolakis]:
Denote:
Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.
A'B'C' = the pedal triangle of P wrt triangle NaNbNc.
La, Lb, Lc = the Euler lines of IBC, ICA, IAB, resp. (concurrent at X(21))
L1, L2, L3 = The parallels to La, Lb, Lc through A', B', C', resp.
Li, Lii, Liii = The reflections of L1, L2, L3 in AI, BI, CI, resp.
1. L1, L2, L3 are concurrent ?
The Euler line of NaNbNc = IN line of ABC (?)
The IN line (?)
Which is the locus of the other parallelogic center (A*B*C*, ABC) as P moves on the IN line?
[César Lozada]:
1) Locus = {Linf} ∪ {Line IN-of-ABC=Euler-line-of NaNbNc}
If IP=t*IN then the point of concurrence Q1(P) lies on the line through ETC’s {11, 113, 942, 1858, 2771, 3028, 3649, 6841, 9955, 11551, 11570, 12047, 12261, 12611, 15904, 22798, 31937} such that
X(11)Q1 = (R*(4*r+(R-2*r)*t)/((3*R+2*r)*r))*X(11)X(942)
ETC pairs : (1,3649), (12,12047), (80,11), (355,6841), (11698,12611), (18357,9955)
NOTE: Line IN pass through ETC’s { 1, 5, 11, 12, 80, 119, 355, 495, 496, 952, 1317, 1387, 1411, 1421, 1483, 1484, 1718, 1807, 1837, 2006, 2594, 2596, 2606, 3614, 4551, 5219, 5252, 5396, 5399, 5400, 5443, 5531, 5533, 5534, 5587, 5660, 5718, 5719, 5720, 5721, 5722, 5723, 5724, 5725, 5726, 5727, 5881, 5886, 5901, 6127, 6264, 6265, 6326, 7173, 7741, 7951, 7958, 7972, 7988, 7989, 7993, 8068, 8070, 8227, 9578, 9581, 9624, 9817, 9897, 10057, 10073, 10283, 10523, 10592, 10593, 10826, 10827, 10886, 10887, 10942, 10943, 10944, 10948, 10949, 10950, 10954, 10955, 10956, 10957, 10958, 10959, 11373, 11374, 11375, 11376, 11698, 11729, 12019, 12025, 12433, 12550, 12735, 12737, 12738, 12739, 12740, 12749, 12750, 12751, 13244, 14204, 14584, 14679, 15017, 15251, 15252, 15253, 15888, 15935, 15943, 15950, 16173, 17602, 17717, 17718, 17719, 17720, 17721, 17722, 17723, 17724, 17725, 17726, 17857, 18357, 19372, 19907, 20586, 22392, 23477, 23513, 23517, 23708, 24217, 24222, 26470, 26475, 26476, 26481, 26482, 31517, 32213, 32214, 32486}
2) Locus: The entire plane.
The parallelogic center ABC->A*B*C* is always Qa=X(35)
For P=x:y:z (barys), the parallelogic center A*B*C*->ABC is:
Q*(P) = a*((2*a-b-c)*(-a^2+b^2+c^2)*x-(2*a^3-(b+c)*a^2-2*(b^2+b*c+c^2)*a+(b^2-c^2)*(b-c))*(y+z)) : :
ETC pairs (P,Q*(P)): (1,2646), (65,35), (950,13411), (1071,6906), (1770,15338), (3057,1), (3555,2975), (3868,3916), (3874,5267), (5697,11011), (6284,12047), (10284,33281), (10572,12), (10914,3871), (10950,10039), , (12672,21740), (12743,8068) , (18412,15837), (21630,214), (24473,17549), (24474,3), (24476,4265), (24680,1385), (24929,24929), (26087,26287)
All lines PQ* pass through T=X(24929) and PQ/QT = -2*(R+r)/R, i.e,
Q=R*P+2*(R+r)*X(24929)
3) The locus such that the given lines concur is {Linf} ∪ {Line through ETC’s {1, 149, 2475, 3120, 5497, 5620, 6224, 11263, 11604, 13146, 14526, 21630, 21907, 26131, 26141, 31019, 33104, 33112, 33134, 33143, 33148, 33155, 33337} } and the locus of the point of intersection Q3 is the line through ETCs {11, 214, 442, 950, 1125, 2646, 6062, 6739, 10543, 10609, 12690, 12743, 17614, 17647, 19889, 25639, 31845, 32557}
If IP=t*IX(149) then X(11)Q3(P) = (2*(r+R*t)/(R+2*r))*X(11)*X(214)
ETC pairs (P, Q3(P)): (1,2646), (21630,214)
RELATED CENTERS:
Q1( X(5) ) = COMPLEMENT OF X(16139)
= (b+c)*a^6-2*b*c*a^5-3*(b^3+c^3)*a^4-2*b^2*c^2*a^3+(b^2-c^2)*(b-c)*(3*b^2+b*c+3*c^2)*a^2+2*(b^2-c^2)^2*b*c*a-(b^2-c^2)^3*(b-c) : : (barys)
= X(3)-3*X(26725), X(21)-3*X(5886), X(79)+5*X(18493), X(191)-5*X(8227), 3*X(1699)+X(16132), X(2475)+3*X(5603), X(3649)+2*X(9955), 2*X(3649)+X(22798), 3*X(5426)-7*X(9624), 3*X(5587)+X(16126), 3*X(5886)+X(16159), 2*X(5901)+X(16125), 2*X(6675)-3*X(11230), 2*X(6701)+X(22791), 4*X(9955)-X(22798), 3*X(11230)-X(22937), 2*X(11544)+X(26202), X(13743)-5*X(18493), X(16113)-3*X(28443), X(16150)+3*X(28453)
= lies on these lines: {1, 18407}, {2, 16139}, {3, 26725}, {5, 758}, {11, 113}, {21, 5886}, {30, 551}, {55, 16155}, {56, 79}, {191, 8227}, {442, 517}, {496, 10122}, {1125, 5428}, {1482, 31140}, {1519, 13373}, {1621, 3651}, {1699, 16132}, {2094, 3652}, {2475, 5603}, {3338, 7701}, {3487, 18517}, {3579, 6690}, {3647, 4999}, {3656, 5330}, {3850, 21635}, {3884, 5499}, {5249, 13624}, {5426, 9624}, {5441, 9668}, {5443, 16142}, {5587, 16126}, {5714, 18516}, {5885, 6831}, {6265, 11604}, {6583, 26470}, {6675, 11230}, {7743, 16193}, {8070, 17605}, {8071, 16153}, {9956, 21677}, {10543, 14526}, {10785, 16116}, {11277, 28174}, {11544, 26202}, {12053, 15174}, {13374, 20288}, {13407, 28204}, {13464, 19907}, {16113, 28443}, {16150, 28453}, {16617, 17768}, {18253, 21616}, {18481, 31019}, {24220, 29097}, {24390, 31938}, {25055, 28460}, {26446, 31254}
= midpoint of X(i) and X(j) for these {i,j}: {21, 16159}, {79, 13743}, {946, 11263}, {3649, 6841}, {3651, 12699}, {3652, 14450}, {3656, 6175}, {5499, 22791}, {6265, 11604}, {16116, 16138}
= reflection of X(i) in X(j) for these (i,j): (1385, 11281), (3647, 10021), (5428, 1125), (5499, 6701), (6841, 9955), (21677, 9956), (22798, 6841), (22936, 16617), (22937, 6675)
= complement of X(16139)
= (K798e)-anticomplement of-X(6675)
= X(6288)-of-3rd Euler triangle
= X(10610)-of-Wasat triangle
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3649, 9955, 22798), (9955, 12047, 12611)
= [ -0.1905574337940544, -0.6225851616571664, 4.1596345632673670 ]
Q1( X(11) ) = MIDPOINT OF X(11) AND X(3649)
= (b+c)*a^6-2*b*c*a^5-(b+c)*(3*b^2-5*b*c+3*c^2)*a^4+(b^2+c^2)*b*c*a^3+(b^2-c^2)*(b-c)*(3*b^2-b*c+3*c^2)*a^2+(b^2-c^2)^2*b*c*a-(b^2-c^2)^3*(b-c) : : (barys)
= X(79)+3*X(16173), 3*X(442)-X(1145), X(1320)+3*X(6175), X(1749)-3*X(3582), X(3647)-3*X(32557), X(3648)-9*X(32558), X(6224)+3*X(11604), 3*X(11263)+X(21630), X(11684)-5*X(31272)
= lies on these lines: {1, 149}, {7, 3065}, {11, 113}, {12, 6797}, {21, 16153}, {30, 1319}, {36, 17009}, {56, 16159}, {79, 104}, {80, 226}, {100, 12609}, {142, 10090}, {191, 499}, {214, 5249}, {354, 1484}, {442, 1145}, {495, 17636}, {496, 13751}, {498, 11024}, {758, 908}, {952, 13407}, {1125, 4996}, {1320, 6175}, {1479, 16132}, {1519, 15528}, {1749, 3218}, {1768, 6847}, {1770, 10058}, {2802, 6701}, {2932, 5880}, {3057, 5499}, {3086, 14450}, {3091, 9803}, {3336, 6972}, {3647, 32557}, {3648, 32558}, {3651, 14798}, {3671, 11571}, {4293, 16118}, {4857, 18444}, {5428, 16142}, {5433, 22937}, {5441, 12053}, {5533, 10122}, {5541, 10056}, {5572, 25558}, {5840, 24299}, {5902, 6830}, {6147, 17660}, {6326, 6826}, {6667, 18253}, {6904, 15015}, {6964, 15017}, {6982, 10051}, {7680, 17654}, {7972, 21620}, {8104, 16151}, {8674, 13158}, {9614, 16143}, {10057, 12531}, {10087, 12855}, {10404, 12773}, {10523, 17665}, {10543, 15178}, {10572, 13273}, {10573, 16126}, {10948, 17653}, {10957, 13852}, {11376, 13743}, {11684, 21616}, {11715, 16125}, {12331, 17718}, {12701, 16117}, {13267, 16147}, {14986, 18224}
= midpoint of X(i) and X(j) for these {i,j}: {11, 3649}, {11715, 16125}
= reflection of X(i) in X(j) for these (i,j): (10122, 18240), (18253, 6667)
= incircle-inverse-of X(5620)
= X(11597)-of-intouch triangle
= X(33565)-of-inverse-in-incircle triangle
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 11263, 14526), (8068, 12736, 1737)
= [ 1.7558732644814440, 2.1217648053792930, 1.3613473407303560 ]
Q1( X(119) ) = MIDPOINT OF X(442) AND X(1537)
= (b+c)*a^9-(b^2+4*b*c+c^2)*a^8-(b+c)*(4*b^2-7*b*c+4*c^2)*a^7+2*(b^2+3*b*c+c^2)*(2*b^2-3*b*c+2*c^2)*a^6+(b^2-c^2)*(b-c)*(6*b^2-5*b*c+6*c^2)*a^5-2*(3*b^6+3*c^6-(b^4+c^4+2*(2*b^2-3*b*c+2*c^2)*b*c)*b*c)*a^4-(b^2-c^2)*(b-c)*(4*b^4+4*c^4-(5*b^2-6*b*c+5*c^2)*b*c)*a^3+2*(b^4-c^4)*(b^2-c^2)*(2*b^2-3*b*c+2*c^2)*a^2+(b^2-c^2)^2*(b-c)^2*(b^3+c^3)*a-(b^2-c^2)^4*(b-c)^2 : : (barys)
= 16*q*p^5-8*(2*q^2+1)*p^4-4*(4*q^2-3)*q*p^3+2*(8*q^4-4*q^2+1)*p^2+4*(q^2-1)*q*p-4*q^2+3 : :, where p=sin(A/2), q=cos((B-C)/2) (trilinears)
= lies on these lines: {11, 113}, {30, 1519}, {119, 7686}, {226, 10742}, {381, 21635}, {392, 5499}, {442, 1537}, {946, 6265}, {1125, 17009}, {2829, 11281}, {5450, 5886}, {6866, 9803}, {7701, 23708}, {12248, 31019}, {12515, 12609}, {14526, 15950}, {16139, 21616}, {16159, 22753}, {20288, 26470}, {22935, 28452}
= midpoint of X(442) and X(1537)
= reflection of X(17009) in X(1125)
= [ -2.1369881320695540, -3.3669351286936250, 6.9579217858043740 ]
Q*( X(2) ) = REFLECTION OF X(3916) IN X(17549)
= a*(6*a^3-3*(b+c)*a^2-2*(3*b^2+2*b*c+3*c^2)*a+(b+c)*(3*b^2-4*b*c+3*c^2)) : : (barys)
= R*X(2)-2*(R+r)*X(24929)
= lies on these lines: {1, 4004}, {2, 3419}, {3, 11520}, {57, 19705}, {72, 3601}, {78, 16418}, {214, 3748}, {228, 19251}, {376, 5758}, {392, 4428}, {519, 2646}, {549, 24299}, {551, 17614}, {936, 17542}, {942, 13587}, {1385, 3241}, {2094, 19708}, {3534, 31164}, {3555, 3612}, {3653, 11240}, {3655, 11239}, {3689, 4669}, {3740, 5426}, {3742, 15015}, {3746, 19524}, {3873, 17502}, {3897, 31145}, {3916, 17549}, {3928, 19704}, {3957, 5126}, {3984, 17571}, {4018, 5217}, {4855, 5439}, {4881, 5049}, {5044, 16858}, {5249, 9945}, {5436, 19536}, {5777, 28461}, {6940, 15178}, {6942, 24680}, {7483, 12437}, {10609, 13405}, {10707, 22935}, {11518, 19537}, {11523, 19535}, {12100, 13151}, {13411, 17530}, {15174, 24982}, {22836, 31165}, {28443, 31837}
= reflection of X(i) in X(j) for these (i,j): (3916, 17549), (17530, 13411)
= {X(3928), X(30282)}-harmonic conjugate of X(19704)
= [ 1.8424693537525790, 2.3055675755216290, 1.1941318432759080 ]
Q*( X(3) ) = MIDPOINT OF X(4) AND X(11015)
= a*(2*a^6-3*(b+c)*a^5-3*(b-c)^2*a^4+6*(b^3+c^3)*a^3-4*(b^2-b*c+c^2)*b*c*a^2-3*(b^4-c^4)*(b-c)*a+(b^2-c^2)^2*(b-c)^2): : (barys)
= R*X(3)-2*(R+r)*X(24929), 3*X(35)-X(40), 3*X(5086)-5*X(5818), 5*X(8227)-3*X(31159)
= lies on these lines: {1, 3}, {4, 11015}, {5, 5440}, {8, 6892}, {20, 5761}, {21, 26878}, {72, 6914}, {78, 3560}, {145, 5770}, {214, 10993}, {443, 10596}, {912, 6906}, {938, 6961}, {944, 11239}, {946, 28452}, {950, 6882}, {1455, 5399}, {1519, 20420}, {1621, 31838}, {1858, 10058}, {1900, 7497}, {3419, 6862}, {3487, 6948}, {3488, 6891}, {3555, 32153}, {3635, 11715}, {3649, 24466}, {3811, 22758}, {3868, 6950}, {3897, 12245}, {4304, 7491}, {4313, 6827}, {4855, 6911}, {5044, 7489}, {5086, 5552}, {5175, 6859}, {5270, 12119}, {5534, 18519}, {5554, 6857}, {5603, 6885}, {5691, 11929}, {5703, 6850}, {5719, 31775}, {5722, 6958}, {5777, 13743}, {5789, 12645}, {5804, 6970}, {5840, 12047}, {5887, 22836}, {5901, 17614}, {6326, 31937}, {6675, 24982}, {6684, 28465}, {6826, 7704}, {6842, 13411}, {6847, 10528}, {6851, 12115}, {6861, 26364}, {6881, 25639}, {6893, 27383}, {6909, 13369}, {6923, 11374}, {6935, 12648}, {6977, 12649}, {7330, 28444}, {7548, 9963}, {8227, 11928}, {8727, 10942}, {9945, 11729}, {9955, 10738}, {10525, 11375}, {10915, 12437}, {14988, 20612}, {15556, 17010}, {16113, 18244}, {16370, 26921}, {17857, 18761}, {31649, 31835}
= midpoint of X(4) and X(11015)
= reflection of X(6842) in X(13411)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10679, 23340), (10202, 23340, 24474), (10284, 11567, 5048)
= [ -1.0101743442892560, -0.5395507764164585, 4.4804339475600440 ]
Q*( X(4) ) = REFLECTION OF X(3916) IN X(3)
= a*(-a^2+b^2+c^2)*(2*a^4-3*(b+c)*a^3-(b-c)^2*a^2+3*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2) : : (barys)
= R*X(4)-2*(R+r)*X(24929), 2*X(946)-3*X(4870)
= lies on these lines: {1, 227}, {2, 355}, {3, 63}, {4, 4313}, {5, 24299}, {8, 6988}, {12, 515}, {20, 5812}, {21, 5777}, {35, 6001}, {36, 12675}, {40, 4421}, {55, 6261}, {65, 6796}, {84, 30282}, {100, 17654}, {104, 6986}, {125, 18641}, {140, 13151}, {214, 6700}, {392, 10267}, {404, 9940}, {405, 5720}, {411, 517}, {474, 18443}, {500, 28381}, {518, 11012}, {535, 4297}, {631, 5768}, {908, 31789}, {936, 958}, {938, 6049}, {942, 6905}, {946, 3058}, {950, 1532}, {952, 6734}, {956, 5534}, {960, 6326}, {971, 6906}, {993, 14872}, {1006, 5044}, {1012, 1490}, {1064, 5266}, {1155, 5884}, {1158, 5217}, {1210, 1319}, {1519, 15171}, {1537, 10624}, {1538, 31795}, {1858, 5172}, {2077, 9943}, {2095, 11520}, {2096, 3522}, {2801, 5267}, {3305, 5780}, {3419, 6825}, {3428, 3811}, {3436, 5731}, {3488, 6848}, {3530, 13226}, {3555, 11249}, {3560, 5927}, {3612, 12114}, {3616, 6864}, {3651, 31793}, {3683, 20117}, {3689, 11362}, {3748, 13464}, {3753, 11499}, {3824, 6901}, {3870, 22770}, {3880, 11014}, {4189, 12528}, {4304, 6260}, {4305, 12667}, {4511, 31786}, {4640, 5693}, {4857, 22835}, {5010, 15071}, {5122, 26877}, {5220, 21153}, {5258, 5531}, {5428, 12691}, {5432, 12616}, {5438, 8726}, {5439, 6911}, {5450, 12680}, {5657, 20007}, {5691, 10894}, {5705, 5881}, {5719, 20420}, {5722, 6834}, {5761, 6869}, {5779, 17571}, {5787, 6833}, {5811, 11111}, {5842, 12047}, {5886, 6835}, {5887, 20846}, {6174, 6684}, {6256, 10953}, {6259, 6938}, {6284, 12608}, {6828, 18480}, {6836, 10526}, {6855, 10585}, {6875, 31445}, {6894, 9955}, {6895, 28160}, {6915, 15178}, {6918, 10246}, {6920, 10157}, {6922, 10942}, {6924, 10202}, {6935, 9799}, {6940, 11227}, {6943, 26287}, {6962, 12649}, {6991, 11230}, {7330, 16370}, {7498, 18283}, {8069, 12711}, {8071, 17625}, {10165, 24953}, {10310, 12520}, {10523, 10572}, {10806, 11373}, {10954, 15844}, {11507, 12709}, {12005, 32636}, {12119, 13272}, {12650, 13384}, {12700, 20075}, {13607, 20323}, {14110, 22836}, {16139, 31663}, {16293, 19861}, {17563, 31657}, {17613, 26285}, {17718, 26332}, {18180, 27653}, {21161, 26878}, {23961, 26201}
= midpoint of X(11491) and X(21740)
= reflection of X(i) in X(j) for these (i,j): (3916, 3), (6831, 13411)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 18446, 1071), (944, 10786, 355), (4855, 10884, 3)
= [ 7.5477567498362470, 7.9958042793978040, -5.3784723652923680 ]
Q3( X(149) ) = REFLECTION OF X(2646) IN X(214)
= a*(2*a^6-3*(b+c)*a^5-(3*b^2-4*b*c+3*c^2)*a^4+(b+c)*(6*b^2-7*b*c+6*c^2)*a^3-4*(b-c)^2*b*c*a^2-(b^3+c^3)*(3*b^2-4*b*c+3*c^2)*a+(b^4-c^4)*(b^2-c^2)) : : (barys)
= X(35)-3*X(15015)
= lies on these lines: {11, 214}, {35, 392}, {72, 2949}, {100, 517}, {149, 24929}, {952, 6734}, {956, 5531}, {960, 14794}, {1145, 6737}, {1319, 33337}, {1385, 5086}, {1484, 24299}, {2771, 3916}, {2802, 11011}, {3555, 22560}, {3582, 10073}, {3897, 20085}, {4855, 11849}, {5289, 11010}, {5541, 11009}, {5687, 11014}, {5886, 6901}, {9945, 11729}, {10087, 17652}, {10090, 13750}, {10914, 12331}
= midpoint of X(i) and X(j) for these {i,j}: {5086, 6224}, {5541, 11009}, {6326, 11012}
= reflection of X(i) in X(j) for these (i,j): (2646, 214), (3916, 4996)
= {X(100), X(22935)}-harmonic conjugate of X(5440)
= [ 1.8046456246447400, -2.1450245495007200, 4.2927681124949980 ]
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